Small Deviation Probabilities Research Articles

Small Deviation Probabilities for Sums of Independent Positive Random Variables

We study the asymptotic behavior at zero of distributions and densities of a sum of several independent positive random variables under certain assumptions on the decay rate of their distributions at zero. We consider cases where the distributions (densities) of summable random variables are regularly or slowly varying at zero or can decrease at zero at an arbitrary rate.

Small deviation probabilities for sums of independent positive random variables

«Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия» — научно-теоретический рецензируемый журнал, публикующий исследования в области математики, механики и астрономии, а также математического моделирования.

On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables

This is the second paper in a series of reviews devoted to the scientific achievements of the Leningrad and St. Petersburg school of probability and mathematical statistics from 1947 to 2017. This paper is devoted to the works on limit theorems for dependent variables (in particular, Markov chains, sequences with mixing properties, and sequences admitting a martingale approximation) and to various aspects of the theory of random processes. We pay particular attention to Gaussian processes, including isoperimetric inequalities, estimates of the probabilities of small deviations in various norms, and the functional law of the iterated logarithm. We present a brief review and bibliography of the works on approximation of random fields with a parameter of growing dimension and probabilistic models of systems of sticky inelastic particles (including laws of large numbers and estimates for the probabilities of large deviations).

Small Deviation Probabilities of a Sum of Independent Positive Random Variables, the Common Distribution of Which Decreases at Zero Not Faster Than Exponential Function

We investigate small deviation probabilities of the cumulative sum of independent positive random variables, the common distribution of which decreases at zero not faster than exponential function.

Small Deviation Probabilities for a Weighted Sum of Independent Positive Random Variables with Common Distribution Function That Can Decrease at Zero Fast Enough

The present note puts forward the most general conditions to date under which the classical asymptotics of the small deviation probabilities for a weighted sum of independent positive random variables takes place. Regarding random variables it is assumed that their common distribution function can, in particular, have an exponential falloff at zero.

‐SMALL DEVIATIONS FOR WEIGHTED STATIONARY PROCESSES

We find logarithmic asymptotics of $L_2$-small deviation probabilities for weighted stationary Gaussian processes (both for real and complex-valued) having power-type discrete or continuous spectrum. As in the recent work by Hong, Lifshits and Nazarov, our results are based on the spectral theory of pseudo-differential operators developed by Birman and Solomyak.

Small Deviation Probabilities for Weighted Sum of Independent Random Variables with a Common Distribution Having a Power Decrease at Zero, under Minimal Moment Assumptions

We represent a certain weight condition such that the exact formula for the asymptotic behavior of small deviation probabilities for weighted sum of independent random variables with a common distribution, which has a power decay at zero, is valid under optimal moment assumptions.

Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional

Nous etudions la mesure de trajectoire transformee engendree par l’auto-interaction d’un mouvement Brownien tridimensionnel en utilisant un biais exponentiel par l’energie de Coulomb des mesures d’occupation de ce mouvement au temps $t$. Les asymptotes logarithmiques de la fonction de partition ont ete identifiees dans les annees 1980 par Donsker et Varadhan [Comm. Pure Appl. Math. 505 (1983) 505–528] au moyen d’une formule variationnelle. Recemment, dans (Brownian occupations measures, compactness and large deviations (2014) Preprint), une nouvelle technique pour etudier la mesure de chemins elle-meme a ete introduite. Elle permet de prouver que la mesure d’occupation normalisee se concentre asymptotiquement autour de l’ensemble des maximums de la formule. Dans le present article, nous prouvons que la fonctionnelle de Coulomb de la mesure d’occupation se concentre elle aussi, dans la topologie uniforme, autour de l’ensemble des fonctionnelles de Coulomb correspondant aux maximums. Ceci represente une etape decisive vers une preuve rigoureuse de la convergence des mesures d’occupation normalisees vers un melange explicite des maximums, derivee dans (Mean-field interaction of Brownian occupation measures, II: A rigorous construction of the Pekar process. Preprint). Nos methodes reposent sur l’obtention de la continuite holderienne de la fonctionnelle de Coulomb de la mesure d’occupation avec des probabilites de deviations exponentiellement petites, en invoquant la theorie des grandes deviations developpee dans (Brownian occupations measures, compactness and large deviations (2014) Preprint) pour une certaine fonctionnelle, invariante par dcalage, des mesures d’occupation.

Small Deviations of Sums of Correlated Stationary Gaussian Sequences

We consider the small deviation probabilities (SDP) in the uniform norm for sums of stationary Gaussian sequences. For the cases of constant boundaries and boundaries tending to zero, we obtain quite general results. For the case of the boundaries tending to infinity, we focus our attention on the discrete analogues of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well-studied FBM case to the discrete time setting under the usual assumptions that imply weak convergence, while the transfer of the corresponding upper bounds necessarily requires a deeper knowledge of the spectral structure of the underlying stationary sequence.

Small deviation probabilities for weighted sum of independent random variables with a common distribution that can decrease at zero fast enough

We examine small deviation probabilities of weighted sum of independent random variables with a common distribution that can decrease at zero fast enough and satisfies mild moment assumptions.

Probabilities of Small Deviations of the Weighted Sum of Independent Random Variables with Common Distribution That Decreases at Zero Not Faster Than a Power

The paper presents estimates of small deviation probabilities of the sum $$ {\displaystyle \sum_{j\ge 1}{\leftthreetimes}_j{X}_j} $$ , where {⋋j} are positive numbers and {Xj} are i.i.d. positive random variables satisfying weak restrictions at zero and infinity. Bibliography: 16 titles.

Small deviations in $L_2$-norm for Gaussian dependent sequences

Let $U=(U_k)_{k\in{\mathbb Z} }$ be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted $\ell _2$-norm small deviation probabilities. It is shown that \[ \ln \mathbb{P} \left ( \sum _{k\in \mathbb{Z} } d_k^2 U_k^2 \le \varepsilon ^2\right ) \sim - M \varepsilon ^{-\frac{2} {2p-1}}, \qquad \textrm{ as } \varepsilon \to 0, \] whenever \[ d_k\sim d_{\pm } |k|^{-p}\quad \textrm{for some } p>\frac{1} {2} \, , \quad k\to \pm \infty , \] using the arguments based on the spectral theory of pseudo-differential operators by M. Birman and M. Solomyak. The constant $M$ reflects the dependence structure of $U$ in a non-trivial way, and marks the difference with the well-studied case of the i.i.d. sequences.

Small Deviations of Probabilities for Weighted Sum of Independent Positive Random Variables with a Common Distribution That Decreases at Zero Not Faster than a Power

In the paper it is shown that the well-known asymptotics for small deviation probabilities of weighed sum of independent positive random variables with a common distribution still holds true if this distribution decreases at zero not faster than some power.

Weighted L p -norms, p ≥ 2, for a wiener process: Exact asymptoties of small deviations

Results concerning exact asymptotics for small deviation probabilities of a Wiener process in Lp-norms with a weight, p ≥ 2, and for Lp-norms of trajectories of some stochastic integrals are proved in the paper.

Small Deviation Probabilities for Sums of Independent Positive Random Variables with a Distribution that Slowly Varies at Zero

In the note, we study small deviation probabilities for sums of independent, identically distributed positive random variables whose distribution function is slowly varying at zero. Bibliography: 5 titles.

Small deviation probabilities of weighted sums under minimal moment assumptions

Abstract We examine small deviation probabilities of weighted sums of i.i.d.r.v. with a power decay at zero under moment assumptions close to necessary.

On small deviation probabilities for certain iterated random processes

The asymptotic behavior of small deviation probabilities for some iterated random processes is investigated. It is shown that, under certain conditions, iterated and noniterated processes have logarithmic asymptotics of the same character; otherwise, these asymptotics may differ substantially. Some iterated Gaussian processes are considered as an example.

Small deviation probabilities for sums of independent positive random variables whose density has a power decay at zero

In this note, we study small deviation probabilities of a sum of i.i.d. positive random variables whose density has a power decay at zero. Bibliography: 3 titles.

Limit theorems for small deviation probabilities of some iterated Stochastic processes

We derive logarithmic asymptotics for probabilities of small deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those of the processes which generate the iterated processes. When these conditions do not hold, the asymptotics of small deviations for iterated processes are quite different. We apply our results to the iterated processes generated by the compound Cox processes and compound renewal processes. Bibliography: 10 titles.

Small deviations of iterated processes in the space of trajectories

AbstractWe derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.